In chapter 1 we look at what was done before Cauchy. Euler and Laplace and others saw that complex changes of variables was a useful technique for evaluating real integrals. This is rather mysterious and it was treated with some suspicion. Cauchy set out to justify and systematise such techniques in his 1814 memoir (chapter 2), and then he kept polishing his results over the next ten years (chapter 3). Then comes his watershed 1825 memoir (chapter 4). Cauchy has now realised that all of the above should be understood in the context of path integration in the complex plane. Here integration is largely determined by the poles, prompting a calculus of residues, which he develops over the next couple of years (chapter 5). Another area of classical analysis where the complex viewpoint proved essential was the convergence of series (chapter 6). Cauchy's starting point here was the Lagrange series, first employed fifty years earlier by Lagrange, e.g. in celestial mechanics, without regard for its dubious convergence properties.
Author(s): Frank Smithies
Edition: 1
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 224
Contents......Page 7
Introduction......Page 9
1 The background to Cauchy's work on complex function theory......Page 14
2 Cauchy's 1814 memoir on definite integrals......Page 32
3 Miscellaneous contributions (1815-1825)......Page 67
4 The 1825 memoir and associated articles......Page 93
5 The calculus of residues......Page 121
6 The Lagrange series and the Turin memoirs......Page 155
7 Summary and conclusions......Page 194
References......Page 213
Notation index......Page 222
Author index......Page 223
Subject index......Page 224
